Currently, tunable diode laser absorption spectroscopy (TDLAS) is a very effective method for measuring parameters such as temperature, component concentration, velocity, or pressure in a combustion flow field. A basic principle of TDLAS is the Beer-Lambert law. By using properties of a laser absorption spectrum of gas molecules, intensity of laser light that penetrates absorbing gas is measured, and a light intensity absorption curve having a particular absorption line shape can be obtained. An absorption line shape function is directly related to field parameters. Values of the parameters can be obtained by performing line shape fitting on a measured absorption curve.
For a gas absorption environment, an absorption line shape is mainly decided by two physical mechanisms: Doppler broadening caused by thermal motion of molecules, and collision broadening caused by a collision between molecules. The two types of broadening respectively produce two corresponding line shape functions: a Gauss line shape function and a Lorentz line shape function. When gas pressure is low, Doppler broadening is dominant, and the absorption line shape is described by using a Gauss line shape. In a high pressure condition, molecules collide more frequently, collision broadening is dominant, and the absorption line shape is described by using a Lorentz shape. Actually, in most cases, there is no great difference between properties of the two types of broadening, and a Voigt line shape needs to be used to describe the absorption line shape. The Voigt line shape is expressed as a convolution of the Gauss line shape and the Lorentz line shape. This convolution expression does not have a definite analytical form. This causes two problems in applying a Voigt line shape function. This first problem is that a long calculation time is required for calculating a value of the Voigt line shape function in an integral form. The second problem is that the Voigt line shape function in the integral form cannot use a conventional line shape fitting tool to perform curve fitting, because a generic line shape fitting tool requires that an analytical expression should be used as an input parameter.
Currently, a general method for solving the two problems is approximating the analytical expression of the Voigt line shape to replace the original integral expression of the Voigt line shape. However, an approximate analytical expression also has its inherent disadvantage, that is, approximation inevitably causes an error. To reduce an error of approximation, an approximate analytical expression with high complexity or even a complex expression needs to be used. However, the complex approximate expression may also cause long-time calculation and non-convergence in line shape fitting. Therefore, for absorption spectrum measurement, a non-approximate Voigt line shape fitting method capable of fast calculation is of great significance.
The following documents and reports relate to calculation and fitting of a Voigt line shape function in laser absorption spectrum measurement.
1. “Rapidly convergent series for high-accuracy calculation of the Voigt function” (Journal of Quantitative Spectroscopy & Radiative Transfer 111 (2010) 372-375), a dissertation by S. M. Abrarov, etc., Yale University, U.S.A. An exponential function sequence based on Fourier expansion is provided for implementing high-accuracy approximation of a Voigt function. Calculation accuracy in this approximate method can reach 10−9. Although calculation in this method is obviously faster than an integral method, an approximate expression is in a form of a sum of sequences and is quite complex.
2. “Implementation of an efficient analytical approximation to the Voigt function for photoemission lineshape analysis” (Journal of Electron Spectroscopy and Related Phenomena 64 (1994) 125-132), a dissertation by A. B. McLean, Queen's University, Canada. A simple approximate expression of a Voigt function is provided. This expression includes a sum of four polynomials in a same form. Each polynomial includes four fixed parameter values. Featuring a simple form and a high calculation speed, this approximate expression is also applicable to line shape fitting, but accuracy of approximation is not high.
3. “Double-peak Fitting of X-ray Diffraction by Voigt Profile Function” (Journal of Synthetic Crystals, issue 2, volume 38, 2009), a dissertation by Zhang Qingli, Anhui Institute of Optics and Fine Mechanics, Chinese Academy of Sciences. A Gauss-Hermite numerical integration formula applicable to M nodes is used for approximating a Voigt line shape function, and a double-peak Voigt profile is fitted by using the approximate expression. A fitting result indicates that a convergence speed and stability thereof are both high. However, this article also points out that an increase of nodes may cause an increase of calculation load, and a calculation speed may be obviously reduced in a case of multiple numerical iterations.
4. “Fast and non-approximate methodology for calculation of wavelength-modulated Voigt lineshape functions suitable for real-time curve fitting” (Journal of Quantitative Spectroscopy & Radiative Transfer 113 (2012) 2049-2057), a dissertation by Jonas Westberg, Umea University, Sweden. A non-approximate method for calculation of wavelength-modulated Voigt line shape functions is provided. In the article, a fast Fourier transform method is used to calculate an expression related to a convolution. The article describes in detail how to obtain an nth-order modulation harmonic factor after Fourier broadening of a wavelength-modulated Voigt line shape function. There is no expression approximation in the whole calculation process, and calculation is fast. However, the article does not describe how to use the method to perform Voigt line shape fitting.
Complexity of accurate calculation of the Voigt line shape function and feasibility of non-approximate calculation are proved in the foregoing documents. However, research on fitting of the Voigt line shape function is still based on an approximate expression. Non-approximate Voigt line shape function calculation makes non-approximate Voigt line shape fitting possible. On this basis, a non-approximate Voigt line shape fitting method for absorption spectrum measurement is implemented in the present invention.